Entanglement entropy of Bell-network states in loop quantum gravity: Analytical and numerical results

Bell-network states are loop-quantum-gravity states that glue quantum polyhedra with entanglement. We present an algorithm and a code that evaluates the reduced density matrix of a Bell-network state and computes its entanglement entropy. In particular, we use our code for simple graphs to study properties of Bell-network states and to show that they are nontypical in the Hilbert space. Moreover, we investigate analytically Bell-network states on arbitrary finite graphs. We develop methods to compute the Rényi entropy of order p for a restriction of the state to an arbitrary region. In the uniform large-spin regime, we determine bounds on the entanglement entropy and show that it obeys an area law. Finally, we discuss the implications of our results for correlations of geometric observables.

Figure 1

Example of a general spin network on a graph $\mathrm{\Gamma }$. We shaded in green a region $A$ which contains $N_A=4$ nodes and $|\partial A|=7$ boundary links (marked with a cross). The region $A$ determines a subsystem ${\mathcal{H}}_A={\otimes }_{n\in A}{\mathcal{H}}_n$.

Figure 2

Graphs and regions considered in our numerical analysis.

Figure 3

[Graph ${\mathrm{\Gamma }}_5$, entropy of a single node]. The figure shows our numerical results on the Bell-network state with pentagram graph ${\mathrm{\Gamma }}_5$ and equal spins $j_{\ell }=\lambda /2$. The entanglement entropy $S_A$ is denoted by blue dots, and the Rényi entropy $R_A^{(2)}$ is denoted by red diamonds. The lower bound asymptotic estimate is shown as a solid orange line [the $O(1)$ is fitted using the numerical data]. The upper bound estimate given by the maximal entropy is shown as a dashed orange line. We show also the bounds (4.20) as a yellow band. The inset shows the 20 data points with the largest spins.

Figure 4

[Graph ${\mathrm{\Gamma }}_5$, entropy of two connected nodes]. The figure shows our numerical results on the Bell-network state with pentagram graph ${\mathrm{\Gamma }}_5$ and equal spins $j_{\ell }=\lambda /2$. The entanglement entropy $S_{AB}$ is denoted by blue dots, and the Rényi entropy $R_{AB}^{(2)}$ is denoted by red diamonds. The lower bound asymptotic estimate is shown as a solid orange line [the $O(1)$ is fitted using the numerical data]. The upper bound estimate given by the maximal entropy is shown as a dashed orange line. We show also the bounds (4.20) as a yellow band. Using the ten largest-spin data points, the entanglement entropy $S_{AB}$ is fitted by $a\log (\lambda )+b+c{\lambda }^{-1}$ on the last ten data points obtaining $a\approx 1.94$, $b\approx -0.30$, and $c\approx 3.19$.

Figure 5

[Graph ${\mathrm{\Gamma }}_6$, entropy of a single node]. The figure shows our numerical results on the Bell-network state with hexagram graph ${\mathrm{\Gamma }}_6$ and equal spins $j_{\ell }=\lambda /2$. The entanglement entropy $S_A$ is denoted by blue dots, and the Rényi entropy $R_A^{(2)}$ is denoted by red diamonds. The lower bound asymptotic estimate is shown as a solid orange line [the $O(1)$ term is fitted using the numerical data]. The upper bound estimate given by the maximal entropy is shown as a dashed orange line. We show also the bounds (4.20) as a yellow band. The inset shows the 20 data points with the largest spins.

Figure 6

[Graph ${\mathrm{\Gamma }}_6$, entropy of two connected nodes]. The figure shows our numerical results on the Bell-network state with hexagram graph ${\mathrm{\Gamma }}_6$ and equal spins $j_{\ell }=\lambda /2$, for a region consisting of two connected nodes. The entanglement entropy $S_{AB}$ is denoted by blue dots, and the Rényi entropy $R_{AB}^{(2)}$ is denoted by red diamonds. The lower bound asymptotic estimate is shown as a solid orange line [the $O(1)$ is fitted using the numerical data]. The upper bound estimate given by the maximal entropy is shown as a dashed orange line. We show also the bounds (4.20) as a yellow band. Using the ten largest-spin data points, the entanglement entropy $S_{AB}$ is fitted by $a\log (\lambda )+b+c{\lambda }^{-1}$ on the last ten data points obtaining $a\approx 1.85$, $b\approx -0.08$, and $c\approx 2.23$.

Figure 7

[Graph ${\mathrm{\Gamma }}_6$, entropy of two disconnected nodes]. The figure shows our numerical results on the Bell-network state with hexagram graph ${\mathrm{\Gamma }}_6$ and equal spins $j_{\ell }=\lambda /2$, for a region consisting of two disconnected nodes. The entanglement entropy $S_{AD}$ is denoted by blue dots, and the Rényi entropy $R_{AD}^{(2)}$ is denoted by red diamonds. The lower bound asymptotic estimate is shown as a solid orange line [the $O(1)$ is fitted using the numerical data]. The upper bound estimate given by the maximal entropy is shown as a dashed orange line. We show also the bounds (4.20) as a yellow band. Using the ten largest-spin data points, the entanglement entropy $S_{AD}$ is fitted by $a\log (\lambda )+b+c{\lambda }^{-1}$ on the last ten data points obtaining $a\approx 1.86$, $b\approx 0.09$, and $c\approx 1.97$.

Figure 8

Diagrammatic representation of $\mathrm{Tr}{M}^2$ for a pentagram graph and a two nodes subsystem.